What if outside the kingdom of Heaven, people who wanted nothing to do with God are actually given a place to live and stay, and they are actually okay?
Well, that is the case, but it depends on what you mean by "okay..."
Existing? Well, yes; check. Happy? Well, no, as evidenced by the rich man in Jesus's parable in Luke 16.
Those outside are not suffering Gods wrath, (I do not believe)...
Well, they will have suffered His wrath, having been resurrected to judgment (John 5:29) and will be "living" under His eternal judgment, which will have been rendered by Jesus at His second coming. They will be "sent away," will "depart," as we see in Matthew 7:23, Matthew 25:41, and Matthew 25:46.
...though there is some type of wonderance of what could have been perhaps?
They will be fully aware of what they could have had, and in anguish/torment because of that, as we see (again) with the rich man of Jesus's parable in Luke 16. This
is their anguish/torment, and thus "their worm that will not die" (Mark 9).
...through the scripture it seems as though God desires for none to perish...
Right, but that does not deter Him from His justice and His glory.
...I really do not believe people are going to be burning in the tormenting hell...
Right, the burning is not literal. Matthew, when we say "trial by fire," none of us is saying that the person is actually being literally burned by fire, right? So, they will exist in this "fire" ~ under God's final judgment, but not literally burning in literal flames.
I also believe if they ever went through the Lake of Fire, there would probably be some suffering there, with God being an all consuming fire.
Right... but, well, not just "some" suffering. The suffering is not really physical, though. See above.
However I do not believe God desires for any to perish...
Well, right, and desires all people to be saved and to come to the knowledge of the truth. This is 1 Timothy 2:4, verbatim. And actually, all
will, eventually, but some... too late, unfortunately.
...so where do people Go when they, whether they had faith or no faith? The Heavenly Realm is presumed.
Well if they had faith, then the "Heavenly Realm," certainly. We all agree on that, although, properly, we would call it the new heaven and new earth, as the Bible does in Revelation 21, right? Now, those with no faith, they will be... well, sent away... they will depart... to... well, all we can really say is, to an absolutely undesirable place.
Grace and peace to you, Matthew.